
is one to one on \(\mathcal {I}_1.\)  It suffices
to build the
relation from the algebro-geometric data \(\lbrace \mu _j(x_0),\nu _j(x_0),E_m;q(x_0),r(x_0)\rbrace _{ j=1,\ldots ,n, m=0,\ldots ,2n+1 }\)  to stationary solutions \(q(x),r(x)\) 
since another direction is obvious.
The Dubrovin-type equations ([1]}, Lemma 3.2)
\(\mu _{j,x}(x) =&\frac{-2iy(\hat{\mu }_j(x))}{\prod _{k=1,k\ne j}^{n}(\mu _j(x)-\mu _k(x))},\ \ j=1,\ldots ,n,\\\nu _{j,x}(x) =&\frac{-2iy(\hat{\mu }_j(x))}{\prod _{k=1,k\ne j}^{n}(\nu _j(x)-\nu _k(x))},\ \ j=1,\ldots ,n,\,\,\) 
